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  • For each place $v$ of $\mathbf{Q}$, define $\Phi_v:(\mathbf{Q}_v)^2\to\mathbf{C}$ by $$ \Phi_v(x,y)=\begin{cases} \mathbb{I}_{\mathbf{Z}_v}(x)\mathbb{I}_{\mathbf{Z}_v}(y)&\text{if $v<\infty$},\\ 2^{-k}(x+\sqrt{-1}y)^ke^{-\pi(x^2+y^2)}&\text{if $v=\infty$}. \end{cases} $$ where $\mathbb{I}_{\mathbf{Z}_v}$ is the characteristic function of $\mathbf{Z}_v$.

  • Define $f_{v,s}:\mathrm{GL}_2(\mathbf{Q}_v)\to\mathbf{C}$ for $s>>0$ by $$ f_{v,s}(g_v)=\lvert{\det(g_v)}\rvert^{s+\frac{1}{2}}\int_{\mathbf{Q}_v^\times}\Phi_v((0,t)g_v)\lvert t_v\rvert^{2s+1}d^\times t_v $$ where the Haar measure are normalized so that $\int_{\mathbf{Q}_v^\times}\mathbb{I}_{\mathbf{Z}_p^\times}(t_v)d^\times t_v=1$. Put $f_s=\bigotimes_vf_{v,s}$, which has analytic continuation to all $s\in\mathbf{C}$.

  • Define the Eisenstein series $\varphi_k:\mathrm{GL}_2(\mathbf{A}_\mathbf{Q})\to\mathbf{C}$ by $\varphi_k(g)=\sum_{\gamma\in B\backslash G}f_s(\gamma g)|_{s=(1-k)/2}$ for positive even integers $k\geq 4$, where $G=\mathrm{GL}_2(\mathbf{Q})$ and $B$ is the subgroup of upper triangular matrices in $G$. I believe that $$ y^{-k/2}\varphi_k\begin{pmatrix}y&x\\0&1\end{pmatrix}=E_k(z):=-\frac{B_{k}}{2 k}+\sum_{n=1}^{\infty} \sigma_{k-1}(n) q^{n},\quad z=x+\sqrt{-1}y $$ which is the classical Eisenstein series of weight $k$. This can be verified by the Fourier expansion $$ \varphi_k(g)=\left.f_s(g)+\sum_{\alpha\in\mathbf{Q}}\prod_v\int_{\mathbf{Q}_v}f_{v,s}\left(\begin{pmatrix}0&-1\\1&0\end{pmatrix}\begin{pmatrix}1&w\\0&1\end{pmatrix}g_v\right)\psi(-\alpha w)dw\right|_{s=(1-k)/2}. $$ However, these are the results of my computations:

  • $f_s\begin{pmatrix}y&x\\0&1\end{pmatrix}|_{s=(1-k)/2}=0$.

  • $\int_{\mathbf{Q}_v}f_{\infty,s}\left(\begin{pmatrix}0&-1\\1&0\end{pmatrix}\begin{pmatrix}1&z\\0&1\end{pmatrix}\begin{pmatrix}y&x\\0&1\end{pmatrix}\right)\psi(-\alpha w)dw|_{s=(1-k)/2}=0$ if $\alpha<0$.

  • $\int_{\mathbf{Q}_v}f_{\infty,s}\left(\begin{pmatrix}0&-1\\1&0\end{pmatrix}\begin{pmatrix}1&z\\0&1\end{pmatrix}\begin{pmatrix}y&x\\0&1\end{pmatrix}\right)\psi(-\alpha w)dw|_{s=(1-k)/2}=y^{k/2}e^{2\pi\sqrt{-1}\alpha z}$ if $\alpha\geq 0$.

For $v<\infty$:

  • $\int_{\mathbf{Q}_v}f_{v,s}\left(\begin{pmatrix}0&-1\\1&0\end{pmatrix}\begin{pmatrix}1&z\\0&1\end{pmatrix}\right)dw|_{s=(1-k)/2}=\zeta(1-k)$.

  • $\int_{\mathbf{Q}_v}f_{v,s}\left(\begin{pmatrix}0&-1\\1&0\end{pmatrix}\begin{pmatrix}1&z\\0&1\end{pmatrix}\right)\psi(-\alpha w)dw|_{s=(1-k)/2}=\sum_{l=0}^{\mathrm{ord}_v\alpha}v^{l(k-1)}$ if $\alpha\neq 0$.

Hence $$ y^{-k/2}\varphi_k\begin{pmatrix}y&x\\0&1\end{pmatrix}=\zeta(1-k)+\sum_{\alpha\in\mathbf{Q}^\times, \alpha>0} e^{2\pi\sqrt{-1}\alpha z}\prod_{v<\infty}\sum_{l=0}^{\mathrm{ord}_v\alpha}v^{l(k-1)}=\color{red}{-\frac{B_k}{k}}+\sum_{n=1}^{\infty}\sigma_{k-1}(n) q^{n}, $$ which is not what I expect. Where is the factor of 2 missing in my calculation? (Or my calculation is correct and $y^{-k/2}\varphi_k\begin{pmatrix}y&x\\0&1\end{pmatrix}\neq E_k(z)$ in fact.)

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